Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [39,3,Mod(7,39)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 11]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39.7");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.l (of order \(12\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(1.06267303101\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.1579585536.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 46\nu^{6} - 150\nu^{5} + 27\nu^{4} + 720\nu^{3} - 1768\nu^{2} + 2792\nu - 2816 ) / 1870 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2702 \nu^{7} - 2868 \nu^{6} + 25735 \nu^{5} - 124331 \nu^{4} + 88530 \nu^{3} + 313004 \nu^{2} + \cdots + 1701898 ) / 753610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3006 \nu^{7} - 12986 \nu^{6} + 9580 \nu^{5} + 30103 \nu^{4} - 124150 \nu^{3} + 454478 \nu^{2} + \cdots + 77946 ) / 753610 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4304 \nu^{7} + 6309 \nu^{6} - 4190 \nu^{5} - 34327 \nu^{4} + 28340 \nu^{3} - 236062 \nu^{2} + \cdots - 1083324 ) / 753610 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 634\nu^{7} - 1964\nu^{6} + 4915\nu^{5} + 902\nu^{4} - 39130\nu^{3} + 104652\nu^{2} - 103278\nu + 63754 ) / 68510 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 14726 \nu^{7} + 49076 \nu^{6} - 12585 \nu^{5} - 244743 \nu^{4} + 585130 \nu^{3} - 751298 \nu^{2} + \cdots - 117106 ) / 753610 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 4\beta_{4} - \beta_{3} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} - 6\beta_{5} + 3\beta_{4} - 2\beta_{2} + \beta _1 - 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} - 6\beta_{6} - 8\beta_{5} + 12\beta_{4} - 10\beta_{3} - 8\beta_{2} - 10\beta _1 + 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - 8\beta_{6} - 36\beta_{5} - 30\beta_{4} - 4\beta_{3} - 24\beta_{2} - 68 \)
|
| \(\nu^{6}\) | \(=\) |
\( 26\beta_{7} + 26\beta_{6} - 22\beta_{5} - 4\beta_{4} - 44\beta_{3} - 70\beta _1 + 48 \)
|
| \(\nu^{7}\) | \(=\) |
\( -44\beta_{7} + 74\beta_{6} - 148\beta_{5} - 272\beta_{4} + 74\beta_{3} + 74\beta_{2} + 22\beta _1 - 316 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−3.09436 | + | 0.829131i | 0.866025 | − | 1.50000i | 5.42349 | − | 3.13125i | 3.29224 | + | 3.29224i | −1.43610 | + | 5.35958i | 9.85686 | + | 2.64114i | −5.12509 | + | 5.12509i | −1.50000 | − | 2.59808i | −12.9170 | − | 7.45766i | ||||||||||||||||||||||||
| 7.2 | 1.72833 | − | 0.463105i | 0.866025 | − | 1.50000i | −0.691437 | + | 0.399201i | 0.707764 | + | 0.707764i | 0.802122 | − | 2.99356i | −2.02673 | − | 0.543062i | −6.07107 | + | 6.07107i | −1.50000 | − | 2.59808i | 1.55102 | + | 0.895482i | |||||||||||||||||||||||||
| 19.1 | −0.247589 | − | 0.924013i | −0.866025 | + | 1.50000i | 2.67160 | − | 1.54245i | 5.21405 | − | 5.21405i | 1.60044 | + | 0.428836i | −2.33731 | + | 8.72296i | −4.79240 | − | 4.79240i | −1.50000 | − | 2.59808i | −6.10879 | − | 3.52691i | |||||||||||||||||||||||||
| 19.2 | 0.613614 | + | 2.29004i | −0.866025 | + | 1.50000i | −1.40365 | + | 0.810399i | −1.21405 | + | 1.21405i | −3.96646 | − | 1.06281i | 1.50718 | − | 5.62489i | 3.98855 | + | 3.98855i | −1.50000 | − | 2.59808i | −3.52518 | − | 2.03527i | |||||||||||||||||||||||||
| 28.1 | −3.09436 | − | 0.829131i | 0.866025 | + | 1.50000i | 5.42349 | + | 3.13125i | 3.29224 | − | 3.29224i | −1.43610 | − | 5.35958i | 9.85686 | − | 2.64114i | −5.12509 | − | 5.12509i | −1.50000 | + | 2.59808i | −12.9170 | + | 7.45766i | |||||||||||||||||||||||||
| 28.2 | 1.72833 | + | 0.463105i | 0.866025 | + | 1.50000i | −0.691437 | − | 0.399201i | 0.707764 | − | 0.707764i | 0.802122 | + | 2.99356i | −2.02673 | + | 0.543062i | −6.07107 | − | 6.07107i | −1.50000 | + | 2.59808i | 1.55102 | − | 0.895482i | |||||||||||||||||||||||||
| 37.1 | −0.247589 | + | 0.924013i | −0.866025 | − | 1.50000i | 2.67160 | + | 1.54245i | 5.21405 | + | 5.21405i | 1.60044 | − | 0.428836i | −2.33731 | − | 8.72296i | −4.79240 | + | 4.79240i | −1.50000 | + | 2.59808i | −6.10879 | + | 3.52691i | |||||||||||||||||||||||||
| 37.2 | 0.613614 | − | 2.29004i | −0.866025 | − | 1.50000i | −1.40365 | − | 0.810399i | −1.21405 | − | 1.21405i | −3.96646 | + | 1.06281i | 1.50718 | + | 5.62489i | 3.98855 | − | 3.98855i | −1.50000 | + | 2.59808i | −3.52518 | + | 2.03527i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 39.3.l.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 117.3.bd.c | 8 | ||
| 13.f | odd | 12 | 1 | inner | 39.3.l.a | ✓ | 8 |
| 39.k | even | 12 | 1 | 117.3.bd.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.3.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 39.3.l.a | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 117.3.bd.c | 8 | 3.b | odd | 2 | 1 | ||
| 117.3.bd.c | 8 | 39.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} - 4T_{2}^{6} + 8T_{2}^{5} + 7T_{2}^{4} - 116T_{2}^{3} + 128T_{2}^{2} - 26T_{2} + 169 \)
acting on \(S_{3}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 169 \)
|
| $3$ |
\( (T^{4} + 3 T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{8} - 16 T^{7} + \cdots + 3481 \)
|
| $7$ |
\( T^{8} - 14 T^{7} + \cdots + 1267876 \)
|
| $11$ |
\( T^{8} + 14 T^{7} + \cdots + 252004 \)
|
| $13$ |
\( T^{8} - 2 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} - 18 T^{7} + \cdots + 1108809 \)
|
| $19$ |
\( T^{8} + \cdots + 136251050884 \)
|
| $23$ |
\( T^{8} + \cdots + 197738523684 \)
|
| $29$ |
\( T^{8} + \cdots + 204578003809 \)
|
| $31$ |
\( T^{8} + \cdots + 557805834496 \)
|
| $37$ |
\( T^{8} + \cdots + 19533696169 \)
|
| $41$ |
\( T^{8} - 22 T^{7} + \cdots + 5650129 \)
|
| $43$ |
\( T^{8} + \cdots + 394461875844 \)
|
| $47$ |
\( T^{8} + \cdots + 119293535376964 \)
|
| $53$ |
\( (T^{4} - 16 T^{3} + \cdots - 1031411)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 2366059240000 \)
|
| $61$ |
\( T^{8} + \cdots + 20246661137641 \)
|
| $67$ |
\( T^{8} + \cdots + 14220787934116 \)
|
| $71$ |
\( T^{8} + \cdots + 4893412955236 \)
|
| $73$ |
\( T^{8} + \cdots + 5324357806849 \)
|
| $79$ |
\( (T^{4} + 40 T^{3} + \cdots + 31352152)^{2} \)
|
| $83$ |
\( T^{8} - 136 T^{7} + \cdots + 418038916 \)
|
| $89$ |
\( T^{8} + \cdots + 23\!\cdots\!16 \)
|
| $97$ |
\( T^{8} + \cdots + 112010726254144 \)
|
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